as \(a\rightarrow +\infty \) . The right-hand side of (REF ) converges only for \(x\in [0,\frac{1}{2})\) , but for any fixed \(x\in [0,1)\)  it is still useful when used as an asymptotic expansion as \(a\rightarrow +\infty \) . For more details, see [1]}. However, it is readily seen that (REF ) breaks down as \(x\rightarrow 1\) . Since this limit has significant importance in applications, Temme derived in [1]} an asymptotic expansion as \(a\rightarrow +\infty \)  that holds uniformly for \(x\in (0,1]\) . His result can be stated as follows.
